Nonlinear Partial Differential Equations in Geometry and Physics

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This volume presents the proceedings of lectures held by the Mathematics Department at The University of Tennessee, Knoxville, from March 22-24, 1995, titled “Nonlinear Partial Differential Equations in Geometry and Physics.” The significance of partial differential equations in differential geometry has been acknowledged since the subject's inception, but their application in formulating physical theories is a more recent development. A pivotal example is the Einstein equations of general relativity, introduced in 1915, which highlight the deep geometric and physical implications of nonlinear partial differential equations. Fundamental aspects of the initial value problem for these equations, such as existence, regularity, and stability of solutions, continue to be vital research areas even eighty years later. Additionally, there is a growing recognition that structures initially developed for theoretical physics models can have substantial geometric or topological applications. This trend may be traced back to 1954, with the introduction of a non-abelian version of Maxwell's equations by physicists C. N. Yang and R. Mills. The intricate geometric structure of the Yang-Mills equations was later emphasized by mathematicians like M. F. Atiyah and J. Hitchin.